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Having disposed of complex numbers by couples, Hamilton
sought to extend his device to ordered triples and quadruples.
Without some idea of what is sought to be accomplished such
an undertaking is of course so vague as to be meaningless.
Hamilton's object was to invent an algebra which would do for
rotations in space of three dimensions what complex numbers,
or his couples, do for rotations in space of fao dimensions, both
spaces being Euclidean as in elementary geometry. Now, a
complex number a -f- bi can be thought of as representing a
vector, that is, a line segment having both length and direction,
as is evident from the diagram, in which the directed segment
(indicated by the arrow) represents the vector OP.


But on attempting to symbolize the behaviour of vectors in
three-dimensional space so as to preserve those properties of
vectors which are of use in physics., particularly in the combina-
tion of rotations, Hamilton was held up for years by an unfore-
seen difficulty whose very nature he for long did not even
suspect. We may glance in passing at one of the clues he
followed. That this led him anywhere - as he insisted it did - is
all the more remarkable as it is now almost universally regarded
as an absurdity, or at best a metaphysical speculation without
foundation in history or in mathematical experience.
Objecting to the purely abstract, postulational formulation